Subsampling estimates of the Lasso distribution.

2.3 A continous mapping theorem for argmin functionals 17
Next, assume without loss of generality that 1 = n(1) < n(2) < . . . and set γ(n) =
max{k : n(k) ≤ n} and define ε n = 1/γ(n). The sequence ε n then satisfies 2.3.0.4
since γ(n) → ∞ and n(γ(n)) ≤ n by definition.
(iii) For i = 1, . . . , k εn , let A n i
and set
⊆ {X n ∈ B (εn)
i } be measurable with
(
P n (A n i ) = P n∗ X n ∈ B (εn) )
A n 0 = Ω n \
k εn
⋃
i=1
A n i .
For each n ∈ N define a probability measure µ n on (Ω n , A n ) as
µ n (A) = 1
k
∑ εn
P n (A|A n i ) ( P n (A n i ) − (1 − ε n ) P ∞ (X ∞ ∈ Bi εn )) .
ε n
i=1
where P n (·|A n i ) is the P n-conditional measure given A n i . Finally, define the probability
space (˜Ω, Ã, ˜P ) as follows :
⎛
⎞
˜Ω = Ω ∞ × ∏ k
∏ εn
⎝Ω n × A n ⎠
i × [0, 1],
n
i=0
⎛
⎞
à = A ∞ × ∏ k
∏ εn
⎝A × A n ∩ A n ⎠
i × B o ,
n i=0
⎛
⎞
˜P = P ∞ × ∏ k
∏ εn
⎝µ n × P n (·|A n i ) ⎠ × λ.
n
Here, B o is the Borel σ-algebra and λ is the Lebesgue measure on [0, 1], respectively.
i=0
(iv) We now define the maps φ n and verify that ˜P ◦ φ −1
n
Define
For A ∈ Ã, we have:
˜ω = ( ω ∞ , . . . , ω, ω n0 , . . . , ω kɛn n, . . . , ξ ) .
i
= P n . Write elements ˜ω of ˜Ω as
φ ∞ = ω ∞
{
ωn , if ξ > 1 − ε n ,
φ n =
ω ni , if ξ ≤ 1 − ε and X ∞ (ω ∞ ) ∈ B (ε)
i .
˜P (φ n ∈ A) = ˜P (φ n ∈ A; ξ > 1 − ε n ) + ˜P (φ n ∈ A; ξ ≤ 1 − ε n )
k εn
= ˜P
∑ (
(ω n ∈ A; ξ > 1 − ε n ) + ˜P ω ni ∈ A; ξ ≤ 1 − ε n ; X ∞ ∈ B (εn) )
i
i=0
k
∑ εn
(
= µ n (A) ε n + P n (A|A n i ) (1 − ε n )P ∞ X ∞ ∈ B (εn) )
i
i=0
k εn
∑
= P n (A|A n i ) P n (A n i )
i=0
= P n (A)